For the next release of GMP, there will be some new public functions for side-channel silent computations, as part of the low-level "mpn" interface.
I think it would make sense to rewrite the RSA and DSA private key operations to use side-channel silent functions. This might require further interface changes; I haven't really looked into it yet, so I don't know to which degree backwards compatibility can be kept.
I also don't yet know if it's practical to make use of the new functions optional, or if it will make the latest GMP a strict requirement (unless Nettle's public key support is completely disabled).
What do you think?
Relatedly, I've updated the release plan on http://www.lysator.liu.se/~nisse/nettle/plan.html.
Regards, /Niels